For these exercises, we'll give whichever answer we think is simplest. If you get something different, consult with a teacher or friend, because you may have a different and correct answer. Like coming up with new nacho topping combinations, there are infinite ways to write the same series.

Example 2

Write the sum using sigma notation.

1 + 1 + 1 + 1 + ...

Answer

The general term is the constant 1. It doesn't matter what value of n we start on, since all the terms are the same. Both

and

are good options.

Example 3

Write the sum using sigma notation.

3 – 6 + 9 – 12 + ...

Answer

We have alternating terms, so we need a factor of (-1) raised to some power.

The general term

an = (-1)n + 13n

makes sense if we start at n = 1. Then the sum looks like

There are other correct answers. For example, we could change the exponent on the (-1) and write

Example 4

Write the sum using sigma notation.

2(0.7)1 + 2(0.7)2 + ... + 2(0.7)25

Answer

The exponents on (.7) range from 1 to 25. We think it makes the most sense to start at n = 1 and write

Example 5

Write the sum using sigma notation.

8 + 10 + 12 + 14 + 16 + ... + 500

Answer

The terms are even numbers starting with

8 = 2(4)

and ending with

500 = 2(250).

A reasonable way to write the sum in sigma notation is

Example 6

Write the sum using sigma notation.

-3 + 0.3 – 0.03 + 0.003 – 0.0003

Answer

Let's deconstruct this series. First, we can pull out a factor of 3 from each term to get

-3 + 3(0.1) – 3(0.01) + 3(0.001) – 3(.0001).

The things in the parentheses are are powers of .1, so write them that way:

-3(0.1)0 + 3(0.1)1 – 3(0.1)2 + 3(0.1)3 – 3(0.1)4

It makes sense to use n = 0 as the starting index, so to get the signs correct we need to have a factor of (-1)n + 1: